Template:Short description In mathematics, a Gaussian rational number is a complex number of the form p + qi, where p and q are both rational numbers. The set of all Gaussian rationals forms the Gaussian rational field, denoted Q(i), obtained by adjoining the imaginary number i to the field of rationals Q.
Properties of the fieldEdit
The field of Gaussian rationals provides an example of an algebraic number field that is both a quadratic field and a cyclotomic field (since i is a 4th root of unity). Like all quadratic fields it is a Galois extension of Q with Galois group cyclic of order two, in this case generated by complex conjugation, and is thus an abelian extension of Q, with conductor 4.<ref>Ian Stewart, David O. Tall, Algebraic Number Theory, Chapman and Hall, 1979, Template:ISBN. Chap.3.</ref>
As with cyclotomic fields more generally, the field of Gaussian rationals is neither ordered nor complete (as a metric space). The Gaussian integers Z[i] form the ring of integers of Q(i). The set of all Gaussian rationals is countably infinite.
The field of Gaussian rationals is also a two-dimensional vector space over Q with natural basis <math>\{1, i\}</math>.
Ford spheresEdit
The concept of Ford circles can be generalized from the rational numbers to the Gaussian rationals, giving Ford spheres. In this construction, the complex numbers are embedded as a plane in a three-dimensional Euclidean space, and for each Gaussian rational point in this plane one constructs a sphere tangent to the plane at that point. For a Gaussian rational represented in lowest terms as <math>p/q</math> (i.e. Template:Tmath and Template:Tmath are relatively prime), the radius of this sphere should be <math>1/2|q|^2</math> where <math>|q|^2 = q \bar q</math> is the squared modulus, and Template:Tmath is the complex conjugate. The resulting spheres are tangent for pairs of Gaussian rationals <math>P/Q</math> and <math>p/q</math> with <math>|Pq-pQ|=1</math>, and otherwise they do not intersect each other.<ref>Template:Citation.</ref><ref>Template:Citation.</ref>